Model order reduction (MOR) is about reducing the computational load in simulation of high dimensional systems, called full order models (FOMs). Examples are discretized partial differential equations with millions of states, e.g., in fluid dynamics or continuum mechanics. A popular geometric intuition for MOR is to think of FOMs as vector fields on differentiable manifolds (the state space, or full order manifold), and to think of reduced order models (ROMs) as vector fields on a submanifold (the reduced order manifold). In this talk I explore an alternative point of view, in which a family of ROMs is described on a low-dimensional Lie group that acts on the full order manifold. I cover abstract theory, promises and challenges of the formulation, and present first examples concerning data-based optimization of Lie groups, their actions, and families of ROMs for point cloud dynamics.